hybrid and adaptive multi
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objective evolutionary algorithm. Soft computing [online], 19(12), pages 3551
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Available from: https://doi.org/10.1007/s00500
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DECMO2: a robust hybrid and adaptive multi
-objective evolutionary algorithm.
ZĂVOIANU, A.
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C., LUGHOFER, E., BRAMERDORFER, G., AMRHEIN, W.
and KLEMENT, E.P.
2015
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DECMO2 - A Robust Hybrid and Adaptive Multi-Objective
Evolutionary Algorithm
Alexandru-Ciprian Z˘avoianu · Edwin Lughofer · Gerd Bramerdorfer ·
Wolfgang Amrhein · Erich Peter Klement
Received: date / Accepted: date
Abstract We describe a hybrid and adaptive coevolu-tionary optimization method that can efficiently solve a wide range of multi-objective optimization problems (MOOPs) as it successfully combines positive traits from three main classes of multi-objective evolutionary algo-rithms (MOEAs): classical approaches that use Pareto-based selection for survival criteria, approaches that rely on differential evolution, and decomposition-based strategies. A key part of our hybrid evolutionary ap-proach lies in the proposed fitness sharing mechanism that is able to smoothly transfer information between the coevolved subpopulations without negatively im-pacting the specific evolutionary process behavior that characterizes each subpopulation. The proposed MOEA also features an adaptive allocation of fitness evalua-tions between the coevolved populaevalua-tions in order to increase robustness and favor the evolutionary search strategy that proves more successful for solving the MOOP at hand. Apart from the new evolutionary algo-rithm, this paper also contains the description of a new hypervolume and racing-based methodology aimed at providing practitioners from the field of multi-objective optimization with a simple means of analyzing/reporting the general comparative run-time performance of multi-objective optimization algorithms over large problem sets.
A.-C. Z˘avoianu·E. Lughofer·E. P. Klement
Department of Knowledge-based Mathematical Sys-tems/Fuzzy Logic Laboratory Linz-Hagenberg, Johannes Kepler University of Linz, Austria
A.-C. Z˘avoianu · G. Bramerdorfer · W. Amrhein · E. P. Klement
LCM, Linz Center of Mechatronics, Linz, Austria G. Bramerdorfer·W. Amrhein
Institute for Electrical Drives and Power Electronics, Johannes Kepler University of Linz, Austria
Keywords evolutionary computation· hybrid
multi-objective optimization · coevolution · adaptive
allocation of fitness evaluations·performance analysis
methodology for MOOPs
1 Introduction
A multi-objective optimization problem (MOOP) can be defined as:
minimizeO(x) = (o1(x), . . . , om(x))T, (1)
wherex∈D,D is called the decision (variable) space,
O : D → Rm consists of m single-objective functions
that need to be minimized andRmis called theobjective
space. In many cases the decision space of the MOOP
is itself multidimensional, e.g.,D=Rn.
Usually, MOOPs do not have a single solution. This
is because the objectives to be minimized (o1. . . om
from (1)) are often conflicting in nature (e.g., cost vs.
quality, risk vs. return on investment) and nox∈D is
able to simultaneously minimize all of them. In order to define a complete solution for a MOOP, we must first
introduce the notions ofPareto dominance andPareto
optimality. When considering two solution candidates
x, y ∈ D, solution x is said to Pareto-dominate
solu-tion y (notation: x y) if and only if oi(x) ≤ oi(y)
for everyi∈ {1, . . . , m} andoj(x)< oj(y) for at least
onej∈ {1, . . . , m}(i.e.,xis better thany with regard
to at least one objective and isn’t worse than y with
regard to any objective). A solution candidatex∗∈D
with the property that there exists noy∈D such that
yx∗is called aPareto optimal solutionto (1). The set
that reunites all the Pareto optimal solutions is called thePareto-optimal set(PS) and this set is the complete
solution of the MOOP. The projection of the Pareto set
on the objective space is called thePareto front (PF).
Since for many problems, the PS is unknown and may contain an infinity of solutions, in real-life
ap-plications, decision makers often use the Pareto
non-dominated set (PN) which contains a fixed number of solution candidates that are able to offer a good ap-proximation of the PS. Therefore, finding high-quality Pareto non-dominated sets is the goal of most multi-objective optimization algorithms (MOOAs). Section 4 contains a detailed discussion regarding quality assess-ment in the case of PNs.
General research tasks in industrial environments
often deal with highly dimensional (6≤n≤60)
multiple-objective (2≤m≤6) optimization problems (MOOPs)
that also may display very lengthy optimization run-times. This is because these industrial optimization sce-narios require fitness evaluation functions that are ex-tremely computationally intensive. For instance, in Yagoubi et al (2011) MOOAs are used for the optimization of combustion in a diesel engine and the fitness evalua-tions require the usage of software emulators. In Jannot et al (2011), finite element simulations are used dur-ing the fitness evaluation of an industrial MOOP from the field of electrical drive design. In these cases, de-spite using modern solving techniques from the field of soft computing like response surface methods, particle swarm optimization, and evolutionary algorithms, for many real-life MOOPs, a single optimization run can take several days, even when distributing the computa-tions over a computer cluster.
As we strive to significantly reduce the run-times required to solve industrial MOOPs, our experience is grounded in three research lines:
– applying non-linear surrogate modeling techniques
on-the fly in order to significantly reduce the de-pendency on computationally intensive fitness
eval-uations (Z˘avoianu et al (2013a));
– deciding what type of parallelization/distribution
method is more likely to deliver the best results tak-ing into consideration the MOOAs that are used and the particularities of the hardware and software
ar-chitecture (Z˘avoianu et al (2013c));
– trying to develop a new MOOA that generally
re-quires fewer fitness evaluations in order to reach an acceptable solution, regardless of the specific MOOP considered, and that is robust with regard to its
pa-rameterization (Z˘avoianu et al (2013b));
While the third research direction is quite general and thus appeals to a considerable larger audience than the former two, it is also, by far, the most challenging. In the present article, building on past findings, we
de-scribe the results of our latest efforts directed towards
developing an efficient and robust multi-objective
op-timization algorithm based on a hybrid and adaptive evolutionary model.
The remainder of this paper is organized as follows: Section 2 contains a short review on multi-objective evolutionary algorithms, Section 3 contains the detailed description of DECMO2, Section 4 presents our ideas on how to perform a general comparison of run-time MOOA performance over large problem sets and a for-mal description of what we understand by the syntagm “robust and efficient” in the context of MOOAs, Section 5 contains a comparative analysis of the performance of DECMO2 versus four other MOOAs when consid-ering a wide range of artificial and real-life MOOPs, and Section 6 concludes the paper with a summary of achievements and some perspectives for future work.
2 Multi-objective evolutionary algorithms
Because of their inherent ability to produce complete Pareto non-dominated sets over single runs, multi-objective evolutionary algorithms (MOEAs) are a particular type of MOOAs that have emerged as one of the most suc-cessful soft computing models for solving MOOPs (Coello et al (2007)).
Among the early (by now, classical) MOEAs, NSGA-II (Deb et al (2002a)) and SPEA2 (Zitzler et al (2002)) proved to be quite effective and are still widely used in various application domains. At a high level of abstrac-tion, both algorithms can be seen as MOOP orientated
implementations of the same paradigm:the(µ+λ)
evo-lutionary strategy. Moreover, both algorithms are highly elitist and make use of similar, two-tier, selection for survival operators that combine Pareto ranking (pri-mary quality measure) and crowding indices (equality discriminant). The names of these Pareto-based
selec-tion for survival operators are: non-dominated sorting
(for NSGA-II) andenvironmental selection(for SPEA2).
Canonically, both NSGA-II and SPEA2 also use the same genetic operators: simulated binary crossover -SBX (Deb and Agrawal (1995)) and polynomial mu-tation - PM (Deb and Goyal (1996)).
More modern MOEAs, like DEMO (Robiˇc and
Fil-ipiˇc (2005)) and GDE3 (Kukkonen and Lampinen (2005))
intend to exploit the very good performance exhibited by differential evolution (DE) operators (see Price et al (1997)) and replaced the SBX and polynomial mutation operators with various DE variants but maintained the elitist Pareto-based selection for survival mechanisms introduced by NSGA-II and SPEA2. Convergence
bench-mark tests (Robiˇc and Filipiˇc (2005); Kukkonen and
help MOEAs to explore the decision space far more ef-ficiently for several classes of MOOPs.
Decomposition is the basic strategy behind many traditional mathematical programming methods for solv-ing MOOPs. The idea is to transform the MOOP (as defined in (1)) into a number of single-objective opti-mization problems, in each of which the objective is an
aggregation of all theoi(x), i∈ {1, . . . , m}, x∈D.
Pro-vided that the aggregation function is well defined, by combining the solutions of these single-objective opti-mization problems, one obtains a Pareto non-dominated set that approximates the solution of the initial MOOP. Miettinen (1999) provides a valuable review of several methods for constructing suitable aggregation functions. However, solving a different single-objective optimiza-tion problem for each soluoptimiza-tion in the PN is quite in-efficient. A major breakthrough was achieved with the introduction of MOEA/D in Zhang and Li (2007) and its DE-based variant (MOEA/D-DE) in Li and Zhang (2009). This evolutionary algorithm decomposes a multi-objective optimization problem into a number of single-objective optimization subproblems that are then si-multaneously optimized. Each subproblem is optimized through means of (restricted) evolutionary computa-tion by only using informacomputa-tion from several of its neigh-boring subproblems. It is noteworthy that MOEA/D proposes a different paradigm to multi-objective opti-mization than most of the previous MOEAs, and that this approach has proven quite successful, especially when dealing with problems with complicated Pareto-optimal sets. A version of MOEA/D (see Zhang et al (2009)) won the CEC2009 Competition dedicated to multi-objective optimization. As such, MOEA/D is con-sidered state-of-the-art by many researchers in the field.
In Z˘avoianu et al (2013b), we described DECMO - a
hybrid multi-objective evolutionary algorithm based on cooperative coevolution that was able to effectively in-corporate the pros of both individual search strategies upon which it was constructed. The idea was to simul-taneously evolve two different subpopulations of equal
size: subpopulation P was evolved using the SPEA2
evolutionary model, while subpopulationQwas evolved
using DEMO/GDE3 principles. After various experi-ments, we discovered that a dual fitness sharing mech-anism is able to induce the most stable behavior and to achieve competitive results. The DECMO fitness shar-ing mechanism consists of:
– generationalweak sharing stages (i.e. trying to
in-sert in each subpopulation one random offspring generated in the complementary subpopulation);
– fixed intervalstrong sharing stages(i.e. constructing
an elite subset of individuals from A=P∪Qand
reinserting this subset inP and Q with the intent
of spreading the best performing individuals across both subpopulations);
The aforementioned elite subset construction and the insertion and reinsertion operations are all performed by applying Pareto-based selection for survival oper-ators (non-dominated sorting or environmental selec-tion).
DECMO can be considered as a successful proof of concept as it displayed a good performance on a benchmark composed of several artificial test problems (the coevolutionary algorithm was consistently able to replicate the behavior of the best performing individual strategy and, for some problems, even surpassed it).
3 Our proposal: DECMO2
In this section, we describe DECMO2, a new and signif-icantly improved variant of our coevolutionary MOEA . Apart from Pareto-based elitism, differential evolution and coevolution, DECMO2 has two more key build-ing blocks (integration of a decomposition strategy and search adaptation) and initial results show that it is able to compete with, and sometimes outperform, state-of-the-art approaches like MOEA/D and GDE3 over a wide range of multi-objective optimization problems.
Like its predecessor, DECMO2 is a hybrid method that uses two coevolved subpopulations of equal and
fixed size. The first one, P (|P| = Psize), is evolved
using the standard SPEA2 evolutionary model. The
second subpopulation, Q(|Q| =Qsize), is evolved
us-ing differential evolution principles. Apart from these,
DECMO2 also makes use of an external archive, A,
maintained according to a decomposition-based strat-egy. The coevolutionary mechanism is redesigned in or-der to allow for an effective combination of all three search strategies and of a search adaptation mechanism. We now proceed to describe the five building blocks of the DECMO2 multi-objective optimization algorithm and, finally, in Section 3.6 we present the algorithmic description of our hybrid evolutionary approach.
3.1 Pareto-based elitism
The cornerstone of the SPEA2 model (used in DECMO2
to evolve subpopulationP) is the environmental
selec-tion (for survival) operator introduced in Zitzler et al (2002). Because we make extensive reference to it, we
shall mark it with Esel(P op, count), with the
under-standing that we refer to the procedure through which
we select a subset of maximumcountindividuals from
an original set P op. The first step is to assign a
of this general rank indicates a higher quality individ-ual. This general rank is the sum of two metrics, the
raw rankr(x) (2) and the densityd(x) (3). In order to
compute the raw rank, each individual x, x ∈ P op is
initially assigned a strength values(x) representing the
number of solutions it Pareto-dominates in P op. The
raw rank assigned to x is computed by summing the
strengths of all the individuals in the population that
Pareto-dominate individualx, i.e.,
r(x) = X
y∈P op:yx
s(y). (2)
The density d(x) of individual x is computed as the
inverse of the distance to the k-th nearest neighbor,
i.e.,
d(x) = 1
distE(x, k) + 2 (3)
wheredistE(x, k) is the Euclidean distance in objective
space between individual xand itsk-th nearest
neigh-bor withk=p|P op|. After each individual inP ophas
been ranked, we simply select the firstcountindividuals
with lowest general rank values. The only noteworthy
detail is that theEsel(P op, count) variant we use across
DECMO2 first removes all duplicate values from P op
and then begins the ranking process.
In DECMO2, at each generationt, t≥1, from the
current subpopulation P, we use binary tournament
selection, SBX and polynomial mutation to create a
new offspring population P0. We then proceed to
con-struct the union of the parent and offspring
popula-tions:P0=P0∪P. Finally, the population of the next
generation is obtained after applying the elitist environ-mental selection operator to extract the best individuals
from this union:P =Esel(P0, Psize)
3.2 Differential evolution
Differential evolution is a global, population-based, stochas-tic optimization method introduced in Storn and Price (1997). By design, DE is especially suitable for continu-ous optimization problems that have real-valued objec-tive functions. Like most evolutionary techniques, DE starts with a random initial population that is then gradually improved by means of selection, mutation and crossover operations.
In the case of DECMO2, at each generationt, t≥1,
subpopulationQwill be evolved using theDE/rand/1/bin
strategy according to an evolutionary model that is
very similar to the ones proposed in DEMO (Robiˇc and
Filipiˇc (2005)) and GDE3 (Kukkonen and Lampinen
(2005)).
At first we perform the initialization: Q0 = Φ and
Q00=Q. Afterwards, as long asQ006=Φ, we randomly
selectx∈Q00 and:
– firstly, we construct the mutant vector v using the
rand/1 part of the DE strategy by randomly
se-lecting three individuals z1, z2, z3 ∈ Q such that
z16=z26=z36=xand then computing:
v=z1+F(z2−z3) (4)
whereF >0 is a control parameter.
– secondly, we generate the trial vector y using the
binomial crossover part of the DE strategy:
yi= (
vi if Ui< CR or i=j
xi if Ui≥CR and i6=j
, (5)
wherejis a randomly chosen integer from{1, . . . , n},
U1, . . . , Un are independent random variable
uni-formly distributed in [0,1], and CR ∈ [0,1] is a
control parameter. n is the dimensionality of the
decision space (D) of the MOOP we wish to solve.
– thirdly, we removexfrom the list of individuals that
we must evolve in the current generation (i.e.,Q00=
Q00\ {x}) and updateQ0: Q0= Q0∪ {x} if xy Q0∪ {y} if yx Q0∪ {x} ∪ {y} if x6y andy6x (6)
At the end of the previously described cycle, it is highly
likely that |Q0|> Q
size because whenxand y are not
dominating each other, both individuals are added toQ0
(i.e., the third case from (6)). In order to obey the fixed subpopulation size design principle, when computing the population of the next generation, we apply the
en-vironmental selection operator (i.e.Q=Esel(Q0, Qsize)).
3.3 Decomposition-based archive
Apart from the two coevolved populations, DECMO2
also uses an archive population,A, that is maintained
according to a decomposition approach that is based on theChebyshev distance.
Let us mark with:
– z∗= (z1∗, . . . , zm∗) the current optimal reference point
of (1). More formally,z∗i =min
oi(x)|x∈DE for each i ∈ {1, . . . , m} when DE ⊂D is the set con-taining all the individuals that have been evaluated during the evolutionary search till the current mo-ment.
– λi = (λi1, . . . , λim), λij ≥ 0 for all j ∈ {1, . . . , m} and m P j=1 λi
j = 1 and i ∈ {1, . . . ,|A|} an arbitrary
objective weight vector;
– dCheb(x, λi) = max
1≤j≤m
λij|oi(x)−zi∗| , x ∈ D the weighted Chebyshev distance between an individual
x∈D and the current optimal reference point;
For any MOOP we wish to solve, we consider a
to-tal of|A|uniformly spread weight vectors:λ1, . . . , λ|A|.
These vectors are generated before the beginning of the evolutionary process and remain constant through-out the entire optimization run. When using them in
thedCheb distance, these weight vectors are the means
through which we define the decomposition of the
origi-nal MOOP problem into a number of|A|single-objective
optimization problems. As such, at any given moment
during the optimization, archive A is organized as a
set of pairs (2-tuples)hλi, yii, yi∈D, where λi is fixed
andyi∈DE has the property that it tries to minimize
dCheb(yi, λi).
Given a current individualxthat has just been
gen-erated during the optimization run, after performing the standard fitness evaluation:
– we update the reference pointz∗;
– we constructA0 - the improvable subset of the
cur-rent archive set:
A0=
yi|∃hλi, yii ∈A:dCheb(x, λi)< dCheb(yi, λi)
(7)
– ifA0 6=Φ, we:
– mark withy∗ that individual inA0 that has the
property thatδCheb=dCheb(y∗, λ∗)−dCheb(x, λ∗)
is maximal (i.e., we apply a greedy selection prin-ciple);
– update the archive by replacing the most
im-provable individual (i.e.,A=A\ hλ∗, y∗i) with
the current individual:A=A∪ hλ∗, xi;
It is worthy to note that the working principles behind the decomposition-based archive are inspired and fairly similar to those proposed by MOGLS (see Jaszkiewicz (2002)) and especially MOEA/D.
3.4 Search Adaptation
By design, nearly all evolutionary models are adap-tive in the sense that, by promoting “a survival of the fittest” strategy, these algorithms are forcing the evolved population to “adapt” with each passing generation (i.e., retain the genetic features that are beneficial for solving the current problem).
The central idea of the DECMO algorithm (Z˘avoianu
et al (2013b)) was to combine the different search be-haviors of classical MOEAs that rely on SBX and PM with that of newer approaches that use DE operators. This was done in light of strong empirical evidence that one evolutionary model was by far better than the other one (when using standard parameterization) on sev-eral well-known problems (i.e., an occurrence subject to the No Free Lunch Theorem by Wolpert and Macready (1997)). By effectively incorporating both search strate-gies, DECMO displayed a good average performance and proved its ability to adapt on a meta level (i.e., to mimic the best strategy for the problem at hand).
In order to improve the aforementioned results, for DECMO2 we designed a mechanism that is aimed to di-rectly bias the coevolutionary process towards the par-ticular search strategy that is more successful during the current part of the run. This is implemented by
dynamically allocating at each odd generation t, t ≥
1 andt ∈ {2k+ 1 :k∈Z} an extra number (Bsize) of
bonus individuals that are to be created and evalu-ated by the search strategy that was able to achieve the highest ratio of archive insertions in the previous (even-numbered) generation. Therefore, at each even generation, we are computing:
– φP - the archive insertion ratio achieved by theP
size offspring generated in subpopulation P via tourna-ment selection, SBX and PM;
– φQ - the archive insertion ratio achieved by the
Qsize offspring generated in subpopulation Q via
DE/rand/1/bin;
– φA- the archive insertion ratio achieved when
creat-ing Bsize offspring by applyingDE/rand/1/bin on
individuals selected directly fromA. When creating
offspring directly fromA, the parent individuals
re-quired by the DE/rand/1/bin strategy are selected
such as to correspond to the Bsize single-objective
optimization problems (i.e., 2-tuples) that have not been updated for the longest periods.
Taking into account previous notation and descriptions,
if, at an arbitrary even generation t, φP > φQ and
φP > φA, at generation t+ 1, the size of the offspring
population (i.e.,P0) will be set to Psize+Bsize.
Like-wise, if, at an arbitrary even generation t, φQ > φP
and φQ > φA, at generation t+ 1, after the stopping
criterion is initially met (i.e.,Q00 =Φ),Q00 will be
re-initialized with a (smaller) set containing Bsize
indi-viduals randomly extracted from Qt and the offspring
generation process will resume until Q00 becomes void
again. If neither of the previous two success conditions
are met, then the Bsize bonus offspring of generation
t+ 1 will be created by applying DE/rand/1/bin on
As we have defined all the individual population subdivision in DECMO2, now, we can also give all the formulae that describe the relation between the size of the archive and the sizes of the two coevolved popula-tions:
|A| =Psize+Qsize+Bsize
Psize =Qsize Bsize = |A| 10 (8) 3.5 Cooperative coevolution
Coevolution is a concept inspired from biological sys-tems where two or more species (that are in a symbi-otic, parasitic or predatory relationship) gradually force each other to adapt (evolve) in order to either increase the efficiency of their symbiotic relationship or survive. For a valuable overview please see Chapter 6 from Luke (2013).
In the field of soft computing, coevolution is usually applied to population-based optimization methods and is implemented using subpopulations that are evolved
simultaneously.N-population cooperative coevolution is
a particular type of coevolutionary process that is mod-eled according to symbiotic relationships occurring in nature. The central idea is to break up complicated
high-dimensional search spaces into N, much simpler,
subspaces that are to be explored by independent (sub)populations. In order to discover high-quality solutions, it is
neces-sary to (occasionally) share information regarding fit-ness between the different populations.
In DECMO2, the particular way in which we apply cooperative coevolution does not implement the previ-ously described search space partitioning concept. In
our case, bothP andQexplore the same search space,
i.e., D. Instead, our approach makes takes full
advan-tage of two general (complementary) characteristics of the coevolutionary concept:
– it helps to maintain diversity in the evolutionary
system;
– it enables the rapid dissemination of elite solutions;
According to the descriptions from the previous four
subsections, at the end of every generationt, t≥1, the
subpopulations P and Q that will be involved in the
next evolutionary cycle (i.e., that of generation t+ 1)
have been computed and archiveA is in an up-to-date
state. The very last step before starting the
computa-tion cycles of generacomputa-tiont+1 consists of a fitness sharing
stage between the three main population subdivisions:
P,Q, andA. The purpose of this stage is to make sure
that the best global solutions found till now are given the chance to be present in both coevolved populations.
The first step is to generateC- an elite subset with
the property:|C|=Bsize, whereBsizehas been defined
in the previous subsection. This elite subset is easily
constructed by first performing the unionC=P∪Q∪A
and then applying the environmental selection
opera-tor:C =Esel(C, Bsize). The second step of the fitness
sharing stage is to try to introduce the individuals of
this elite subset into the subpopulations P and Q of
the next generation. This is also done through the us-age of the environmental selection operator (as defined in Section 3.1):
– P =P∪C andP =Esel(P, Psize);
– Q=Q∪C andQ=Esel(Q, Qsize);
3.6 The main DECMO2 loop
The initialization stage (i.e., “generation 0”) and the main computational loop of our hybrid MOEA are pre-sented in Algorithm 1. There are three input
param-eters: M OOP - the definition of the problem to be
solved, archS - the size of the archive A (i.e., |A|),
andmaxT - the maximum number of generations to be
evolved. The algorithm returns aPareto non-dominated
set (PN) of sizearchS.
There are seven auxiliary methods that we use across Algorithm 1:
– ExtractSizes(archS) - this function computesPsize,
Qsize, andBsizefrom the call argumentarchS(which
equals |A|), by solving (8);
– InitializeArchive(M OOP, archS) - considering the notations from Section 3.3, this function first
creates a total of archS uniformly spread weight
vectors (i.e.,λ1, . . . , λarchS) with the dimensionality
required by the given M OOP. It then proceeds to
create and return an incomplete archive of the form:
A=
hλ1,i, . . . ,hλarchS,i ;
– CreateIndividual(M OOP) - this function returns a randomly created individual that encodes a
pos-sible solution for the givenM OOP;
– InsertIntoArchive(A, x) - this procedure looks if there are any incomplete pairs (i.e., of the form
hλi,iwithi∈ {1, . . . ,|A|}) in archiveA, and, if such
a pair is found, it updates the archive:A=A\ hλi,i
andA=A∪ hλi, xi;
– EvolveNextGenSPEA2(P, Psize) - this function uses the SPEA2 evolutionary model described in Section 3.1 in order to evolve the solution set passed
as the first call argument (i.e.,P) for one generation.
It returns two entities: i) a new, evolved, population
Algorithm 1 Description of the DECMO2 hybrid multi-objective evolutionary algorithm
1: function DECMO2(M OOP, archS, maxT) 2: P, Q←Φ
3: hPsize, Qsize, Bsizei ←ExtractSizes(archS)
4: A←InitializeArchive(M OOP, archS) 5: i←1 6: whilei≤archS do 7: x←CreateIndividual(M OOP) 8: InsertIntoArchive(A,x) 9: if i≤Psizethen 10: P←P∪ {x} 11: else
12: if i≤Psize+Qsize andi > Psizethen
13: Q←Q∪ {x} 14: end if 15: end if 16: i←i+ 1 17: end while 18: φP, φQ, φA←1 19: t←1 20: whilet6=maxT do 21: if t∈ {2k+ 1 :k∈Z}then 22: if φP > φQandφP > φA then 23: Psize=|P|+Bsize 24: end if 25: if φQ> φP andφQ> φAthen 26: Qsize=|Q|+Bsize 27: end if 28: end if
29: Asize←archS−Psize−Qsize
30: hP, φPi ←EvolveNextGenSPEA2(P, P size) 31: hQ, φQi ←EvolveNextGenDE(Q, Q size) 32: φA←EvolveArchiveInd(A, A size) 33: Psize=|P| 34: Qsize=|Q| 35: C←P∪Q∪A 36: C←Esel(C, Bsize) 37: P←P∪C 38: P←Esel(P, Psize) 39: Q←Q∪C 40: Q←Esel(Q, Qsize) 41: t←t+ 1 42: end while 43: C←P∪Q∪A 44: C←Esel(C, archS) 45: returnC 46: end function
by thePsize offspring that were created during the
evolutionary process.
– EvolveNextGenDE(Q, Qsize) - this function uses the DE-based evolutionary cycle described in Sec-tion 3.2 in order to evolve the set passed as the first
call argument (i.e.,Q) for one generation. It also
re-turns two entities: i) a new, evolved, population of
size|Q| and ii) the archive insertion ratio achieved
by theQsizeoffspring that were created during the
evolutionary cycle.
– EvolveArchiveInd(A, Asize) - this function uses theDE/rand/1/bin strategy (i.e., the combination
of (4) and (5)) to create a number of Asize
off-spring using only individuals directly selected from
the pairs that make up archive A. Each offspring
individual created at this step is considered for the archive update procedure described in Section 3.3. The archive insertion ratio achieved when
consider-ing the Asize generated offspring is the only entity
returned by this function. IfAsize= 0, the function
returns the value 0.
When considering the above description, one of the major shortcomings of our proposed multi-objective op-timization method is evident: high structural and com-putational complexity. As we strived to create an effi-cient hybrid starting from three different evolutionary approaches for solving MOOPs, ending up with a fairly complex optimization procedure was something to be expected. However, it should be noted that, apart from the parameterizations required by the genetic opera-tors we rely on (i.e., SBX, PM and DE), our approach remains quite robust, as it does not require any extra parameters. In Section 5 we present solid evidence that DECMO2 displays a very good average performance on a wide range of MOOPs and we think that this more than compensates for the complexity of our method.
4 Comparing the performance of MOOAs
As with most (meta)heuristic approaches, when talking about the performance of a multi-objective evolution-ary algorithm, three criteria are primarily considered and usually need to be balanced:
– thequality of the generated solution, i.e., how well does the PN returned at the end of the optimization run approximate the PF of the MOOP to be solved?
– theconvergence speed, i.e., what is the number of
fit-ness evaluations (notation: nf e) that must be
per-formed during the optimization run in order to reach
a PN ofacceptable quality?
– thegeneralityof the algorithm, i.e., is the proposed method able to display the previous two criteria on a wide range of problems?
It should be noted that the above three criteria can be applied to evaluate any multi-objective optimization algorithm. For example, very fine-grained grid searches over the entire decision space will likely produce the best results with regard to the quality and generality criteria, but such approaches display excessively poor convergence speeds, which render them useless in most cases.
Over the years, several metrics for assessing the PN quality criterion have been proposed. A comprehensive
analysis and review of most of these metrics can be found in Zitzler et al (2003). Some of the more pop-ular metrics are: the generational distance (GD) and the inverted generational distance (IGD) proposed in Van Veldhuizen and Lamont (1998) and the
hypervol-ume metric (H) proposed in Zitzler (1999). The latter
has the added advantage that it is the only PN qual-ity comparison metric for which we have theoretical proof of a monotonic behavior (see Fleischer (2003)). As such, by design, the PF of any MOOP has the highest
achievable Hvalue. The monotonic property of Hcan
be understood in the sense that, given two Pareto
non-dominates sets,P NAandP NB, ifH(P NA)>H(P NB),
we can be certain thatP NA “is not worse than”P NB
(see Zitzler et al (2003) for details). Furthermore, when comparing with GD and IGD, the hypervolume is eas-ier to compute when the PF of the MOOP is unknown (as it is the case with most real-life problems).
Measuring the convergence speed is a truly trivial
task once one has a clear idea of how to defineacceptable
qualityin the case of Pareto non-dominated sets. Unfor-tunately this definition is highly domain-dependent and sometimes it also depends on the experience (or even subjective opinions) of the decision maker. For example, in many publication from the field of multi-objective
op-timization,acceptable quality means a (nearly) perfect
approximation of the PF of a given benchmark MOOP. When considering real-life applications of multi-objective
optimization, a PN may be deemed of having an
accept-able quality if it “is not worse than” any other PN ever discovered for the considered MOOP (even though it is actually a rather poor approximation of the PF).
In light of the very computationally intensive na-ture of the fitness functions required by the industrial MOOPs we aim to solve, our idea of how to best bal-ance quality, convergence speed and generality in order to assess the performance of a MOOA is that: given an arbitrary MOOP and a maximal number of fitness
evaluations (nf emax) that we are willing to execute,
the analyzed MOEA displays the best possible
perfor-mance if, for anynf e≤nf emax, the PN obtained after
performing nf efitness evaluations “is not worse than”
the PN that might have been obtained by any another
available method after also performingnf efitness
eval-uations.
Although quite vague at a first glance, the previ-ous statement is the base from which we developed a practical ranking framework for multi-objective opti-mization algorithms. This framework is described in the next subsection and it can offer practitioners valuable
insight regarding therelative performance of different:
– multi-objective optimization algorithms;
– parameterization settings for a given MOOA;
4.1 A racing-based ranking of performance in the context of MOOPs 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500
True hypervol. (avg. over 4 MOOPs) [%]
nfe [in hundreds]
Averaged run-time hypervolume performance
Alg-A Alg-B Alg-C Alg-D
Fig. 2 Averaged run-timeH-measured performance over the entire toy test set
Let us consider a toy example in which we wish to compare the performance of four different multi-objective optimization algorithms (A, B, Alg-C, and Alg-D) on a limited test set that consists of four benchmark MOOPs (P1, P2, P3, and P4) with known PFs. For each optimization run we perform 50000 fit-ness evaluations. A more or less standard approach would be to perform several independent runs for each MOOA-MOOP pair and assess the quality and convergence be-havior by computing some metric over averaged results. For example, the plots from Figure 1 display the
aver-age run-timeH-measured performance of the four
algo-rithms when considering 25 independent runs for each test. For every run, the data points were obtained by
calculating theHof the current MOOA population
af-ter every 100 fitness evaluations and afaf-terwards
com-puting the percentage ratio (notation:H −%−ratio)
obtained when comparing these values againstH(P F),
whereP F denotes the Pareto front of the MOOP that
is solved. As a side note, we shall write “fully” (with quotes) when referring to a MOOA that is able to solve
a MOOP (i.e.,H −%−ratio≈100%) in order to
em-phasize the fact that, in most cases (e.g., all continuous MOOPs), a PN cannot be (by definition) more than a near perfect approximation of the PF.
In order to quickly assess the general performance of the four tested MOOAs (w.r.t. the example test set),
it is very intuitive to plot theH −%−ratio-measured
performance, averaged over the entire problem set(e.g., Figure 2). Such a chart is very useful as it clearly shows:
– which algorithm generally starts to converge faster
0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500 True hypervolume [%]
nfe [in hundreds]
Test problem P1 Alg-A Alg-B Alg-C Alg-D 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500 True hypervolume [%]
nfe [in hundreds]
Test problem P2 Alg-A Alg-B Alg-C Alg-D 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500 True hypervolume [%]
nfe [in hundreds]
Test problem P3 Alg-A Alg-B Alg-C Alg-D 0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500 True hypervolume [%]
nfe [in hundreds]
Test problem P4
Alg-A Alg-B Alg-C Alg-D
Fig. 1 Run-timeH-measured performance on the four problems considered in our toy test set
0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 µS ranks [-] Ranking stage nr. [-]
Basic ranking schema
Alg-A Alg-B Alg-C Alg-D 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 µS ranks [-] Ranking stage nr. [-]
Pessimistic ranking schema
Alg-A Alg-B Alg-C Alg-D
Fig. 3 HRPCs obtained when applying the racing-based ranking methodology on the toy test set
– which algorithm has the best average performance
at the end of the runs (e.g., Alg-C);
– which algorithms seem to have a somewhat
simi-lar convergence behavior during (a certain part of) the optimization run. For example, Alg-D and Alg-B
converge quite fast (averageH−%−ratio≥70%
af-ter 15000 fitness evaluations) while A and Alg-C converge slower, but reach slightly better average results at the end of the experiment (after 50000
fit-ness evaluations). Knowing this and assuming that the used MOOP test set is relevant for real-life sit-uations, in practice, we would prefer Alg-D or even Alg-B over Alg-C/Alg-A when faced with a limited number of fitness evaluations and we would use Alg-C or Alg-A if there would be no such limitation.
Nevertheless, the average H plot from Figure 2 is
also misleading because, through averaging, it helps to mask very bad performance. As all four algorithms
dis-Table 1 Ranks corresponding to the run-timeHplots presented in Figure 1. For each algorithms, the highlighted values are used to create the left-side plot from Figure 3.
Rank computation stages based onH −%−ratios
Problem 0 1 2 3 4 5 6 7 8 9 10 µP
Ranks achieved by Alg-A
P1 5 5 5 5 4 4 3 3 3 2 2 3.73 P2 5 1 1 1 1 1 1 1 1 1 0 1.27 P3 5 2 1 1 1 1 1 1 1 2 2 1.64 P4 5 5 5 5 4 4 4 4 4 4 4 4.36 µS 5.00 3.25 3.00 3.00 2.50 2.50 2.25 2.25 2.25 2.25 2.00 µA= 2.75,µF = 2.00
Ranks achieved by Alg-B
P1 5 5 5 2 2 2 2 2 2 3 3 3.00 P2 5 4 4 4 4 4 4 4 4 4 4 4.09 P3 5 5 4 4 4 4 4 4 4 4 4 4.18 P4 5 1 0 0 0 0 0 0 0 0 0 0.55 µS 5.00 3.75 3.25 2.50 2.50 2.50 2.50 2.50 2.50 2.75 2.75 µA= 2.95,µF = 2.75
Ranks achieved by Alg-C
P1 5 1 1 3 3 3 4 4 4 4 4 3.27 P2 5 5 3 3 3 3 3 2 2 2 2 2.82 P3 5 5 3 3 3 3 3 3 3 1 1 3.00 P4 5 5 3 3 3 3 3 0 0 0 0 2.27 µS 5.00 3.50 2.50 3.00 3.00 3.00 3.25 2.25 2.25 1.75 1.75 µA= 2.84,µF = 1.75
Ranks achieved by Alg-D
P1 5 5 2 1 1 1 1 1 1 1 1 1.82 P2 5 2 2 2 2 2 2 3 3 3 3 2.64 P3 5 1 2 2 2 2 2 2 2 3 3 2.36 P4 5 2 0 0 0 0 0 0 0 0 0 0.64 µS 5.00 2.5 1.5 1.25 1.25 1.25 1.25 1.5 1.5 1.75 1.75 µA= 1.86,µF = 1.75
play average H −%−ratio values between 85% and
95% after 40000 fitness evaluations, we might believe that their general performance is largely similar when it comes to the solutions discovered towards the end of each run. In fact, the very good performance of Alg-B on P1, P2 and P4 helps to cover up the very poor be-havior on problem P3. Similarly, the fact that all four algorithms are (sooner or later) each able to fully con-verge on one MOOP is also concealed. Although in our very simple example, both problems can be solved by independently consulting the relative performance of the four MOOAs on each MOOP via numerical/visual
inspection ofH-related performance, in rigorous
perfor-mance comparison contexts, involving tens of MOOPs and several MOOAs, such a case-by-case approach is very tedious, and, in the late stages of convergence (where most good algorithms find PNs of roughly sim-ilar quality), it can also become useless.
Our idea for simplifying the comparison process is to interpret the run-time hypervolume plot for each
MOOP as if it depicts the results of amulti-stage race
between the four MOOAs. The goal is to reach aH −
%−ratio≈100 as fast as possible (i.e., “fully” converge
after the lowest possiblenf e). The secondary goals are
to have the highestH−%−ratioat the end of each stage
in the race. Therefore, it makes sense to imagine abasic
ranking schemawhere, at the end of each stage, the an-alyzed MOOAs are ranked in ascending order of their
H −%−ratiostarting with the worst performer. In our
toy example, 4 is assigned for the smallestH−%−ratio
value and 1 for the highest. There are two exceptions from this rule:
– if theH −%−ratioat a certain stage is higher than
99% (i.e. the obtained PN dominates more than 99% of the objective space that is dominated by the PF), the analyzed algorithm is assigned the rank 0. This is how we mark (reward) “full” convergence.
– if the H −% −ratio at a certain stage is lower
than 1%, the analyzed algorithm is assigned a rank which equals one plus the total number of analyzed MOOAs (i.e., 5 in our case). This is how we mark (penalize) a MOOA that has not yet produced a relevant PN, i.e., a MOOA that has not started to converge.
In the toy example, the comparison stages are equidis-tant (i.e., they are set after every 5000 fitness evalua-tions) and, in Figure 1, each stage of the race is marked with a vertical solid grey line . The rank information we obtained is presented in Table 1. For each MOOA, the table also presents four average ranks:
– µP - the average rank achieved by the MOOA on
an individual problem. A value closer to 0 indicates that the algorithm displays a good performance on
the problem .µPcan be used to rapidly/automatically
identify those problems on which a MOOA performs very well (i.e., “fully” converges very fast) or poorly (i.e., does not “fully” converge, converges very slowly, etc.).
– µS - is the average rank across the entire test set
at a given stage (i.e., after a fixed number ofnf es).
These are useful as we shall combine them in or-der to display the dynamics of the relative MOOA performance over time.
– µF - is the average rank across the entire test in the
final stage (i.e., close to the end of the optimization).
The MOOA that has the smallest value ofµF was
able to “fully” converge or discover higher quality PNs on more problems than its competitors.
– µA - is the overall average rank achieved by the
MOOA during the comparison. The value ofµAcan
be used to single out the MOOAs that tend to gen-erally outperform their counterparts.
In the left-side plot from Figure 3 we use the µS
values to plot hypervolume-ranked performance curves
(HRPCs). We feel that by introducing HRPCs, we are providing practitioners in the field of multi-objective optimization with an extremely useful tool for helping to rapidly assess the general comparative performance of MOEAs (especially over test sets containing many
MOOPs). The basic ranking schema ignores the
mag-nitudes of the differences in performance and favors the algorithm that is able to perform very well on the high-est number of MOOPs from the considered thigh-est set. When considering the HRPCs computed for the toy comparison context (left-side plot from Figure 3), the data points corresponding to the last 2 ranking stages indicate that:
– there is a good balance between the number of MOOPs
on which Alg-D and Alg-C perform well by the end of the optimization runs;
– Alg-A has managed to converge on at least one MOOP
right before the final stage as passing from a rank of 1 to a rank of 0 is the only explanation for a drop of average rank between the two stages that does not influence the average ranks of the other three MOOAs;
The main advantage of HRPCs is that they can be easily adjusted in order to outline certain MOOA performance characteristics by making small changes in the required ranking procedure. For example, us-ing the same run-time information that was plotted in Figure 1, we could focus our MOOA comparison on analyzing if there are large differences in performance between the tested algorithms by imposing that: at a
given stage, the difference between twoH −%−ratios
must be higher than 10% in order to justify a rank
im-provement, i.e., we impose aH −rankingthreshold of
10% According to the this modification, if at a
cer-tain stage the four MOOAs have the H −%−ratios
(64%,78%,84%,99.5%), they will be assigned the ranks
(4,3,3,0). The HRPCs obtained when applying this,
very pessimistic, ranking schema are presented in the right-side plot from Figure 3 and:
– the data points corresponding to the last 3 ranking
stages confirm that Alg-D and Alg-C have an aver-age similar convergence behavior towards the end of the runs;
– the data points corresponding to the last ranking
stage indicate that Alg-A seems to perform much
worse (i.e., difference inH −%−ratio >10%) than
the other 3 MOOAs on at least one extra MOOP; We have devised this racing and hypervolume-based ranking methodology that may combine information from:
– several HRPC plots (computed using different
rank-ing schemata);
– associatedµP,µF, µAvalues;
– the plot of the averaged H-measured performance
over the entire problem test
in order to easily observe/report the general perfor-mance characteristics of the MOOAs we wish to analyze over large problem sets.
4.2 On robustness and efficiency in MOEAs
As mentioned in the introductory part, our main in-terests with regard to multi-objective optimization al-gorithms are related to enhancing these methods in or-der to improve the run-times of industrial optimizations that rely on very computationally intensive fitness eval-uation functions. In the particular case of MOOAs that can be parameterized (e.g., most MOEAs), the pro-hibitive optimization run-times that occur when solv-ing industrial MOOPs usually make systematic param-eter tuning approaches virtually impossible. This means that we strongly prefer to rely on MOEAs that are very
robust with regard to their control parameters, mean-ing that they generally perform well on a wide range of problems when using the parameterization settings recommended in literature.
The second important characteristic that we demand
from a MOEA isefficiency. In non-academic terms, the
idiom “bang for the buck” encapsulates very well the essence of this characteristic and, in light of the
con-cepts presented till this point, we consider that “Hfor
nf e” is a good equivalent, especially when dealing with
very lengthy run-times induced by computationally in-tensive fitness evaluation functions. The only condition
is that, in the case of MOEAs, efficiency must be
sta-ble(i.e., displayed throughout the duration of the
opti-mization run), general (i.e., displayed on a wide range
of MOOPs), and, because of the stochastic nature of
evolutionary methods, must be supported by averaged
results over many independent runs.
The new MOOA racing-based ranking comparison framework, which we introduced in the previous sub-section, is able to offer insights with regard to both the efficiency and robustness of a given MOEA provided that we construct comparison contexts where:
– we compare the given MOEA against MOOAs that
are themselves regarded as being generally success-ful (i.e., they are state-of-the-art);
– we maintain a fixed parameterization of the tested
algorithms;
– the test sets contain a sufficient number of MOOPs
with different characteristics;
– we apply appropriate ranking schemata;
In the next section, we obey these rules in order to con-struct comparison contexts that help to tune MOEA/D-DE and to evaluate the robustness and efficiency of DECMO2.
5 Tests regarding the performance of DECMO2
In order to evaluate the performance of DECMO2, we consider two types of comparisons:
– the first one aims to estimate the robustness and
effi-ciency of our hybrid and adaptive MOEA by apply-ing the new comparison methodology we proposed in Section 4 on a test set consisting of 20 artificial benchmark problems;
– the second comparison is a case study regarding
the convergence behavior of DECMO2 and SPEA2 on two industrial MOOPs from the field of elec-trical drive design optimization. Both problems re-quire very computationally intensive fitness evalua-tion funcevalua-tions.
The 20 artificial benchmark problems we aggregated in our test set are:
– DTLZ1, DTLZ2, DTLZ4, DTLZ6, and DTLZ7 from
the problem set proposed in Deb et al (2002b);
– KSW10 - a classic optimization problem with 10
variables and two objectives based on Kursawe’s function described in Kursawe (1991);
– all nine problems from the LZ09 problem set
de-scribed in Li and Zhang (2009);
– WFG1, WFG4 and WFG8 from the problem set
proposed in Huband et al (2005);
– ZDT3 and ZDT6 from the problem set described in
Zitzler et al (2000);
When applying the race-based ranking methodol-ogy, we defined ranking stages after every 1000 fitness evaluations with the first ranking evaluation taking place at “generation 0” (i.e., we evaluated the randomly gen-erated initial population of the MOEAs). We performed 50 independent runs for each MOEA-MOOP pair in or-der to obtain the hypervolume information based on which the rankings were computed. We applied two types of ranking schemata:
– the basic ranking schema which is identical to the one described in Section 4.1;
– the pess-T hr ranking schema which has the same
working principles as thepessimistic ranking schema
presented in Section 4.1. T hr is the H −ranking
threshold. For example, a pess-5 ranking schema
uses aH −rankingthreshold of 5%.
The algorithms we compared DECMO2 against (us-ing the race-based rank(us-ing methodology) are SPEA2, GDE3, MOEA/D-DE (the Zhang et al (2009) version) , and DECMO. In the case of the first three algorithms we relied on implementations available in the jMetal package (see Durillo and Nebro (2011)). We fixed the number of fitness evaluations to 50000. Across all runs we used MOEA parameterizations that are in accor-dance with those recommended in literature. For SPEA2,
we used a population and archive size of 200, 0.9 for the
crossover probability and 20 for the crossover
distribu-tion index of SBX, 1/n for the mutation probability
(wherenis the number of variables of the MOOP to be
solved) and 20 for the mutation distribution index of PM. For GDE3, we used a population size of 200 and
the settings CR=0.3 and F=0.5 for theDE/rand/1/bin
strategy. For MOEA/D-DE, we used a population size of 500 and all other parameters were set as described in Zhang et al (2009). For DECMO we used a size of 100 for each coevolved subpopulation, the same SBX and PM parameterizations used for SPEA2 and the settings
CR=0.2 and F=0.5 for theDE/rand/1/binstrategy. In
2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Basic ranking schema
MOEA/D-DE-100 MOEA/D-DE-200 MOEA/D-DE-300 MOEA/D-DE-400 MOEA/D-DE-500 MOEA/D-DE-600 MOEA/D-DE-700 MOEA/D-DE-800 MOEA/D-DE-900 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-5 ranking schema
MOEA/D-DE-400 MOEA/D-DE-500
Fig. 4 HRPCs obtained when testing the impact of various archive sizes on MOEA/D-DE
– in the case of subpopulationP: 1.0 for the crossover
probability and 20 for the crossover distribution
in-dex of SBX, 1/n for the mutation probability and
20 for the mutation distribution index of PM;
– in the case of subpopulationQ: the settings CR=0.2
and F=0.5 for theDE/rand/1/bin strategy;
– in case of (the bonus) individuals evolved directly
from A: the settings CR=1.0 and F=0.5 for the
DE/rand/1/bin strategy;
In the case of DECMO and DECMO2, the
con-trol parameters for theDE/rand/1/binstrategy used in
subpopulationQare chosen such as to maintain a good
trade between exploration and intensification (F=0.5) and, as shown in Zaharie (2009), stimulate a minor in-crease in population diversity (CR=0.2). When
evolv-ing bonus individuals fromA, CR is set to 1.0 in order
to stimulate population diversity to the maximum in-side the highly elitist archive.
While it is fair that all MOEAs except MOEAD/D-DE should use the same population size since they are all constructed around the Pareto-based elitism paradigm, in the case of MOEAD/D-DE, the size of the archive was set at 500 after using the race-based ranking method-ology to estimate the relative performance achieved by nine different archive sizes (from 100 to 900). We
ap-plied thebasic ranking schemaand obtained the HRPCs
that are presented in the left-side plot from Figure 4. These HRPCs indicate that, on average, over the 20 considered benchmark MOOPs:
– when using an archive size of 500, MOEAD/D-DE
is able to achieve the best results towards the end of the optimization run ((i.e., between the ranking stages 31 and 50));
– when using an archive size of 400, MOEAD/D-DE
is able to achieve the best results during the middle of the run (i.e., between the ranking stages 8 and 30);
Having two strong candidates, we applied again the
race-based ranking methodology (this time using a
pess-5 ranking schema) on only MOEAD/D-DE-400 and MOEAD/D-DE-500. The obtained HRPCs are presented in the right-side plot from Figure 4 and they indicate that, on average, differences between the two methods are greater during the end of the run than during the middle part of the run. As such, we decided that, when keeping every other parameter fixed, an archive size of 500 would enable MOEA/D-DE to achieve the best overall performance on our benchmark problem set.
5.1 Results on artificial benchmark problems
0 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 350 400 450 500
True hypervol. (avg. over 20 MOOPs) [%]
nfe [in hundreds]
Averaged run-time hypervolume performance
SPEA2 GDE3 MOEAD/D-DE DECMO2 DECMO
Fig. 5 Averaged run-timeH-measured performance over 20 artificial benchmark MOOPs
Because in several future statements we shall use the phrase “on average” to refer to conclusions drawn from various results we present, it is important to clearly state what we mean by this. Considering that we have experimented with 5 different MOEAs over 20 differ-ent MOOPs and that we made 50 independdiffer-ent runs for
1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Basic ranking schema SPEA2 GDE3 MOEAD/D-DE DECMO DECMO2 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-1 ranking schema SPEA2 GDE3 MOEAD/D-DE DECMO DECMO2 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-5 ranking schema SPEA2 GDE3 MOEAD/D-DE DECMO DECMO2 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-10 ranking schema SPEA2
GDE3 MOEAD/D-DE DECMO DECMO2
Fig. 6 HRPCs obtained when comparing DECMO2 with four other MOEAs over 20 artificial benchmark MOOPs
each MOEA-MOOP combination, at every stage of our race-based ranking procedure we have assigned ranks based on 2000 (when comparing only two MOEAs) to 5000 (when comparing all five) hypervolume measure-ments. Since the HRPCs are based on 51 ranking stages, each of them aggregates information from 102000 (plots with two RHPCs) to 255000 independent hypervolume measurements. In order to construct the plot of the
av-eraged H-measured performance of all five algorithms
over the entire benchmark problem set (i.e., the plot
from Figure 5), we sampledHvalues after 1000 fitness
evaluations on each independent run. Therefore, this plot is based on 2500000 independent data points.
Table 2 The average ranks achieved by the five tested MOEAs over the benchmark problem set when applying a
pess-1 ranking schema. The best values are highlighted.
Algorithm µF µA SPEA2 3.6500 3.8265 GDE3 3.5000 3.7490 MOEA/D-DE 2.6500 3.4775 DECMO 2.9000 2.8902 DECMO2 2.1500 2.3294
Figure 6 contains four subplots with the HRPCs ob-tained by the five algorithms we tested with over the en-tire artificial problem set. In addition, Table 2 presents
theµF andµA values achieved by each tested MOEA
when applying thepess-1 ranking schema. With regard
to “full” convergence (i.e., reaching H −%−ratio >
99%), SPEA2 was able to achieve it on 3 problems, GDE3 on 4 problems, MOEA/D-DE and DECMO on 5 problems, and DECMO2 on 7 problems. LZ09-F1 is the only MOOP on which DECMO2 was unable to achieve “full” convergence, but another algorithm, namely MOEA/D-DE, managed to do so.
Taking into account the setup of our tests and the arguments from Section 4.2, all the previously
men-tioned results allow us to conclude thatDECMO2 is
an efficient and robust MOEA.
Although all the HRPCs from Figure 6 and all the hypervolume average values plotted in Figure 5 show that, on average (and at every stage of the run), our method produces better hypervolumes than the other MOEAs we have compared against, it is extremely im-portant to interpret this information in combination
with the implications of the monotonicity of theH
0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Basic ranking schema MOEAD/D-DE DECMO2 0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-1 ranking schema MOEAD/D-DE DECMO2 0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-5 ranking schema MOEAD/D-DE DECMO2 0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-10 ranking schema MOEAD/D-DE
DECMO2
Fig. 7 HRPCs obtained when comparing DECMO2 with MOEA/D-DE over 20 artificial benchmark MOOPs
the strongest statement that we can make based on the obtained results is that: on average, DECMO2 is not worst than any of the other four MOEAs during any stage of the optimization run. But, based on the pre-sented results, the same statement cannot be made for any of the other four algorithms. In light of this, (for the
considered comparison context/test settings)we can we
can weakly argue that, on average, DECMO2 is the best choice among the five tested MOEAs.
In accordance with the previous line of arguments, and taking into account the HRPCs obtained with the
pess-5 and pess-10 ranking schemata, we can also con-clude that, on average, especially in the initial phases of the optimization runs, DECMO2 displays a conver-gence speed that is not outperformed by any of the other MOEAs. We believe that this feature makes our hybrid algorithm a very strong candidate for MOOPs where the solver is limited in the number of fitness eval-uations that it can perform per optimization run.
In Figure 7 we plot the HRPCs obtained when only comparing DECMO2 to MOEA/D-DE. They indicate clearly that, on average, our hybrid and adaptive MOEA displays a better convergence behavior during the early part of the run and that MOEA/D-DE is, more or less,
able to generally match the performance of DECMO2 towards the end of the run.
In Figure 8 we plot the HRPCs obtained when only comparing DECMO2 to DECMO. The HRPCs
corre-sponding to thebasic and pess-1 ranking schemata
in-dicate that, in comparison with its predecessor, on av-erage, DECMO2 displays at least some small improve-ments throughout the entire optimization run. When
applying thepess-5 andpess-10 ranking schemata, DECMO2
only shows an improved average performance during the early part of the run and a slightly better average performance towards the end of the run. Nevertheless, the general idea is that by adding a decomposition-based strategy and an adaptive allocation of fitness evaluations, we were able to increase the overall perfor-mance of our initial coevolutionary method and enable it to successfully compete with a very well known and successful multi-objective optimizer like MOEA/D-DE over a wide range of artificial MOOPs.
5.2 Case study: electrical drive design
Although extremely valuable for the algorithm design, prototyping and parameter tuning stages, as we are
0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Basic ranking schema DECMO DECMO2 0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-1 ranking schema DECMO DECMO2 0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-5 ranking schema DECMO DECMO2 0 1 2 3 0 5 10 15 20 25 30 35 40 45 50 µS ranks [-] Ranking stage nr. [-]
Pess-10 ranking schema DECMO DECMO2
Fig. 8 HRPCs obtained when comparing DECMO2 with DECMO over 20 artificial benchmark MOOPs
primarily motivated by practical applications of multi-objective optimization, we generally regard the assess-ment of MOOA performance on artificial MOOPs as a mere means to an end. The final objective is to ob-tain a robust MOOA that is able to successfully tackle real-life MOOPs.
Using the same parameterizations we experimented with on the artificial problem set, we applied DECMO2 on two fairly complicated MOOPs from the field of elec-trical drive design, allowing for 10000 fitness evaluations per run. In both problems, the goal is to configure 22 real-valued parameters in order to simultaneously op-timize 4 objectives regarding cost and efficiency. For each problem, in order to evaluate the quality of a sin-gle design, we must perform a series of computation-ally intensive operations consisting of a meshing stage and one or more finite element simulations. The overall impact of these required simulations is that, even when distributing the fitness evaluations over a high through-put cluster comthrough-puting environment, performing 10000 fitness evaluations takes between 6 and 7 days to com-plete.
Because of the extremely long run-times, we only performed two independent runs with DECMO2 for
each problem and saved information regarding the best found solutions after every 100 fitness evaluations. For both industrial MOOPs we also have (historical) run-time quality information from optimizations conducted with SPEA2 (two independent runs for each MOOP). Using as reference the best known sets of solutions for both problems, in Figure 9 we present the run-time
H-measured performance of DECMO2 and SPEA2 on
the two industrial problems. The results indicate that DECMO2 is able to converge faster. In this particular case, the faster convergence of DECMO2 roughly
trans-lates into finding PNs that have the sameH values as
those that were discovered one day later when using SPEA2.
6 Conclusion
In this paper, we have described DECMO2, a hybrid multi-objective optimization algorithm that uses co-evolution to successfully combine three different princi-ples for solving MOOPs: Pareto-based dominance, dif-ferential evolution and decomposition-based strategies. DECMO2 also incorporates an adaptive allocation of fitness evaluations in order to accelerate convergence
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 True hypervolume [%]
nfe [in hundreds]
Industrial MOOP nr. 1 SPEA2 DECMO2 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 True hypervolume [%]
nfe [in hundreds]
Industrial MOOP nr. 2
SPEA2 DECMO2
Fig. 9 Run-timeH-measured performance of DECMO2 and SPEA2 on two industrial MOOPs
by rewarding the incorporated evolutionary model that is able to deliver the best performance during a given part of the optimization run.
A considerable part of the present paper (Section 4.1) is dedicated to introducing a new methodology aimed at providing practitioners from the field of multi-objective optimization with a simple means of analyz-ing/reporting the general comparative run-time perfor-mance of MOOAs over large problem sets. This method-ology is largely based on a racing perspective over aver-aged hypervolume measures and can be used either to fine tune algorithms over given problem sets or to
an-alyze the relative robustness and efficiency of MOOAs
(see the discussion from Section 4.2).
In Section 5 we present results using the newly intro-duced MOOA comparison methodology that substan-tiates the claim that DECMO2 displays both robust-ness and efficiency when comparing against four other MOEAs (SPEA2, GDE3, MOEA/D-DE and DECMO) over a challenging benchmark of 20 artificial MOOPs from different well known problem sets. The results section also contains a small case study regarding the comparative performance of DECMO2 and SPEA2 on two real-life industrial MOOPs that feature
computa-tionally intensive fitness evaluation functions. The re-sults of this study confirm the general characteristic of DECMO2 to converge fast.
In light of all the presented results, we finally ar-gue that DECMO2 is a valuable addition to the ever-growing set of MOEAs and that, despite its structural complexity, this hybrid evolutionary algorithm is very robust with regard to its parameterization and, there-fore, especially suited for solving real-life MOOPs that have computationally intensive fitness evaluation func-tions.
With regard to DECMO2, future work will revolve around developing a steady-state asynchronous version of the algorithm and around testing and analyzing the comparative performance on more industrial MOOPs. We also plan to extend our racing-based MOOA com-parison methodology by designing a ranking schema that uses statistical significance testing.
Acknowledgements This work was conducted within LCM GmbH as a part of the COMET K2 program of the Austrian government. The COMET K2 projects at LCM are kindly supported by the Austrian and Upper Austrian governments and the participating scientific partners. The authors thank all involved partners for their support.
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